An ideal $I$ is pure when the quotient is a flat module. Equivalently every $a\in I$ satisfies $a=ab$ for some $b\in I$. The latter condition makes $I$ vanish after localizing at primes containing it, while at primes not containing it the quotient is zero. Pure ideals need not be finitely generated. They describe flat morphisms that are closed immersions on affine charts.

An ideal in a polynomial ring generated by monomials. Its exponent vectors form an upward-closed subset of ${\mathbb{N}}^r$ under the componentwise order. The Dickson lemma makes the minimal exponent set finite. This description permits Maclagan's theorem to compare ideals by reverse inclusion independently of the coefficient field.

Ice-shelf water has acquired characteristic properties through interaction with the underside of an ice shelf. Melting at depth can cool and freshen the water, including cooling below the surface freezing temperature because pressure lowers the freezing point. Its eventual density and contribution to deep-water formation depend on mixing with surrounding shelf and ocean waters.

Ice-shelf buttressing is the restraint that a floating ice shelf, through its contact with sidewalls or pinning points, can exert on upstream grounded ice flow. Thinning or loss of the shelf can reduce this restraint and increase discharge, but the response depends on geometry and cannot be inferred from floating-ice area alone.

Sea-ice free drift neglects internal stress transmitted from neighboring ice. An isolated floe is governed by air and water drag forces, the Coriolis force and sea-surface tilt; steady drift also neglects acceleration.

// Target: geophysics.bigb

If $xyz$ is a hyperedge and the links of $x,y,z$ on four other vertices contain the three perfect matchings of a $K_4$, then those seven vertices contain a Fano plane. There is a second completion: if the link of $v$ contains three disjoint pairs, and the other six vertices contain all transversal triples through those pairs, take four triples of even parity and the three triples containing $v$. This again gives a Fano plane. These finite templates convert forbidden Fano copies into missing cross-part triples.

The hypergraph chromatic number is the minimum number of colours in a proper hypergraph colouring. If $H$ has $N$ vertices and maximum hypergraph independent set size $a$, then $\chi (H)\ge N/a$, since the colour classes partition its vertex set into hypergraph independent sets.

A hypergraph has Property B if its vertices of a hypergraph can be coloured with two colours so that every hypergraph edge meets both colour classes. An $r$-uniform hypergraph with fewer than $2^{r-1}$ edges has Property B, by a fair independent colouring and the union bound. Conversely, for $r\ge 2$, take $N=2r^2$ vertices and independently sample $m=\lceil (N\log 2+1)2^r\rceil$ uniformly random $r$-subsets. In any fixed colouring an edge is monochromatic with probability at least $2^{-r}$: convexity of binomial coefficients reduces to the balanced colouring, and $\left(\genfrac{}{}{0ex}{}{r^2}{r}\right)/\left(\genfrac{}{}{0ex}{}{2r^2}{r}\right)\ge 2^{-r-1}$. The union bound over $2^N$ colourings gives a probability at most $2^N{e}^{-m{2}^{-r}}<1$ of any proper colouring. Removing repeated edges leaves a hypergraph without Property B and with $O(r^2{2}^r)$ edges.

The chromatic index of a hypergraph is the minimum number of colours needed to colour its edges so that intersecting edges have different colours. Each colour class is a matching in a hypergraph. It is at least the maximum hypergraph vertex degree, and, when the edge set is nonempty, at most $r(\Delta -1)+1$ for an $r$-uniform hypergraph of maximum degree $\Delta \ge 1$, by greedily colouring edges: an edge meets at most $r(\Delta -1)$ others.

A strain energy density measures stored elastic energy per unit volume. The volume convention must be specified. For a hyperelastic material, write $W(\mathbf{F})$ per unit volume of the reference configuration, where $\mathbf{F}$ is the deformation gradient. The total stored elastic energy iswith $\mathbf{P}$ the first Piola-Kirchhoff stress tensor. If $J=\det \mathbf{F}>0$, the density per current volume is $w=W/J$. At controlled temperature, the corresponding isothermal description uses a suitable free energy.

In uniform small shear strain $\gamma$, a linear elastic material with shear modulus $G$ stores $W=G{\gamma }^2/2$. Differentiating with respect to $\gamma$ gives the shear stress $G\gamma$. Integrating over the specimen gives the work recoverable on elastic unloading. The same construction turns a local material law into a problem in the calculus of variations for a bent or stretched body, subject to its boundary conditions.

For an incompressible neo-Hookean solid, a convenient strain energy density is $W=\mu (\mathrm{tr}C-3)/2$, where $C$ is the right Cauchy-Green deformation tensor. An additive constant in $W$ does not affect the stress. The shear modulus $\mu$ fixes the small-strain stiffness.

An incompressible Mooney-Rivlin solid has strain energy density $W=C_1(I_1-3)+C_2(I_2-3)$, with right Cauchy-Green deformation tensor $C=F^{T}F$, $I_1=\mathrm{tr}C$ and $I_2=\mathrm{tr}{C}^{-1}$ when $\det C=1$. It extends the neo-Hookean solid by a second invariant. A convention $W={\mu }_1(I_1-3)/2-{\mu }_2(I_2-3)/2$ means $C_1={\mu }_1/2$, $C_2=-{\mu }_2/2$; the minus sign must be retained when comparing material constants.

For a regular material constraint $F(A)=0$, admissible rates of the deformation gradient lie in the kernel of $F_{,A}$. The difference between the nominal stress and the derivative of the strain energy density annihilates that tangent space by virtual work, so it is a Lagrange multiplier times $F_{,A}$. For incompressibility, the derivative of the determinant is its cofactor matrix, and the material-first reaction is $q{A}^{-1}$ when $\det A=1$.

For unconstrained hyperelastic material, the Hessian of the strain energy density gives $\delta N_{\alpha i}=c_{\alpha i\beta j}\delta A_{j\beta }$. Commuting the second derivatives gives major symmetry $c_{\alpha i\beta j}=c_{\beta j\alpha i}$. These material-coordinate moduli linearize about a finite base deformation; they need not have the minor symmetries of a small-strain elastic stiffness tensor. Material constraints require their additional reaction increments and admissibility equations.

At leading thin-shell order, a spherical balloon with reference radius $a_0$, thickness $h_0$ and current radius $a=\lambda a_0$ has reference elastic energy $4\pi a_0^2{h}_{0}W(\lambda ,\lambda ,{\lambda }^{-2})$. Equating its differential to pressure work $p4\pi a^{2}da$ gives the displayed pressure difference. Its thickness becomes $h_0{\lambda }^{-2}$ by incompressibility. The same result follows from the spherical membrane force balance $p=2h{\sigma }_t/a$, with current tangential Cauchy stress tensor component ${\sigma }_t=\lambda N$ and the equibiaxial nominal tension of an incompressible sheet. Thin-shell validity requires $h/a\ll 1$.

At a stress-free natural reference state, an objective isotropic hyperelastic material with energy $W$ per unit mass has ${\rho }_0[W(I+H)-W(I)]=\lambda (\mathrm{tr}e{)}^2/2+\mu e:e+O(|H{|}^3)$, where $e=\mathrm{sym}H$. Twice differentiating gives the displayed Lamé parameters and the usual elastic stiffness tensor. A prestressed reference requires initial-stress corrections.

This objective isotropic constitutive model uses the Green-Lagrange strain tensor $E=(A^{\mathsf{T}}A-I)/2$. It reproduces linear elasticity at small strain. Under simple shear $A=[(1,q,0{)}^{\mathsf{T}},e_2,e_3]$, its longitudinal nominal stress is $(\lambda +2\mu )q^2/2$. Nonuniform finite shear therefore generates longitudinal acceleration, even though the corresponding infinitesimal transverse mode is uncoupled.

For a plane material-coordinate elastic wave with $u=(0,w(X_1,t),0)$, set $q=w_{,X_1}$. The longitudinal equation is $0={\partial }_{X_1}{P}_{11}(1,q,0)$. Arbitrary nonuniform pure transverse profiles thus require the displayed constitutive compatibility, which does not follow from isotropy alone. If it holds and ${\tau }^{\prime }(q)>0$, the shear equation closes as ${\rho }_0{w}_{tt}={\partial }_{X_1}\tau (q)$ with speed $c(q{)}^2={\tau }^{\prime }(q)/{\rho }_0$.

The composition of hyperbolic reflections in two ultraparallel hyperbolic lines has infinite order. Map their common perpendicular to the imaginary axis in the Poincaré half-plane model. The two reflecting hyperbolic lines become semicircles centred at zero, with distinct radii $r_1,r_2$. Reflection in the radius-$r$ semicircle is $z\mapsto r^2/\overline{z}$, so their composition is $z\mapsto (r_2/r_1{)}^{2}z$. Every positive power remains nonidentity, because $(r_2/r_1{)}^{2k}\ne 1$ for $k\ge 1$.

A reflection in the perpendicular bisector of a point and its image corrects that point. A second bisector reflection through the corrected point can correct another point without moving the first. An isometry fixing two distinct points is either the identity or reflection in their joining geodesic: points with prescribed distances from both centres have at most two possibilities, exchanged by that reflection. Fixing a third noncollinear point then fixes every point. This constructs a factorization with at most three hyperbolic reflections.